Theoretical Simulation of Resonant

Inelastic X-Ray Scattering In

Transition Metal Oxides

Siobhan Grayson

Senior Sophister Physics

Supervisor: Dr C. McGuinness

Department of Physics

Trinity College Dublin

January 2012

Abstract

Metal L-edge 2p-3d resonant inelastic x-ray scattering (RIXS) spectra of TiO2 was

simulated and compared with previously obtained experimental RIXS spectra. This was

done by applying an atomic multiplet model to the 2p63dN2p53dN+1 transition in

tetragonal symmetry (D4h) and allowing for solid state effects such as crystal field

interactions with the oxygen ligands of the metal ion to be taken into account. X-ray

absorption spectroscopy (XAS) calculations were computed for Ti4+ and Ti3+

configurations in both octahedral and tetragonal symmetries. These calculations

included not only crystal field interactions but also charge-transfer between the metal

ion and surrounding ligands. Ti4+ simulations were compared against a rutile TiO2 Ti

L-edge XAS spectrum. Ti3+ simulations were compared against a sputtered rutile TiO2

Ti L-edge XAS spectrum.

i

Acknowledgements

I would like to express my thanks and sincere gratitude to my supervisor, Dr. Cormac

McGuinness, for his kind support, advice and encouragement. For always making the

time to answer questions, and for being so friendly and approachable. Turning to

Cormac for advice was never a problem and a solution was always found.

I would also like to thank Declan Cockburn, Martin Duignan, and Stephan Callaghan,

whom were never to busy to offer their advice. In particular, my sincere thanks to

Declan for sharing not only his experimental rutile TiO2 XAS spectrum but for his

immense wisdom of Origin and for advising me on how best to present my simulations.

Thanks to Stephan, whom not only shared his sputterd rutile TiO2 XAS spectrum but

also offered me great support when times got tough.

I will miss researching with the TCD X-Ray Group, it’s been a privilege getting to know

you all.

ii

Contents

1 Introduction 1

1.1 X-Ray Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 X-Ray Emission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Resonant Inelastic X-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . 4

1.4 Titanium Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theoretical Backgroud 6

2.1 Atomic Multiplet Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Ligand Field Multiplet Theory . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Crystal Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Charge Transfer Multiplet Theory . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Resonant Inelastic X-Ray Scattering . . . . . . . . . . . . . . . . . . . . . 14

2.6 Spectroscopic Simulation Software . . . . . . . . . . . . . . . . . . . . . . . 14

3 Computational Method 15

3.1 Method for Simulating XAS . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Method for Simulating RXIS . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Results and Discussion 18

4.1 Simulated Ti4+ 2p XAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Octahedral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.2 Tetragonal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Simulated Ti3+ 2p-3d XAS . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Octahedral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.2 Tetragonal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Ti3+ 2p-3d RIXS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Conclusions 29

Appendices 30

5.1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.1 The Dipole Selection Rules . . . . . . . . . . . . . . . . . . . . . . . 30

iii

Contents

5.1.2 The Hartree-Fock Method . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.3 Lamor Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 Broadening Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3.1 Varying the Crystal Field . . . . . . . . . . . . . . . . . . . . . . . 35

5.3.2 Crystal Field Parameters . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3.3 RIXS Simulation: Incident Energy Vs Emitted Energy . . . . . . . 40

Bibliography 40

iv

Chapter 1

Introduction

Resonant inelastic x-ray scattering (RIXS) belongs to a family of experimental techniques

known as spectroscopy. Spectroscopy is the study of the interaction between matter

and radiated energy. It is a broad field encompassing many sub-disciplines, each with

numerous implementations of specific spectroscopic techniques. These techniques are a

powerful tool for probing the electronic structure of materials, which characterise most of

a material’s relevant attributes. Due to the recent advent of high-brilliance synchrotron

radiation sources [1], the development of one field in particular, known as core level

spectroscopy, has been remarkable. Core level spectroscopy is used to study valence

electron states (VES), the states responsible for the bonding in molecules and solids [2].

Such photon energies belong mainly to the soft x-ray region (between 100 eV and

3 keV). The VES are detected with the core electron or (core hole) state as a probe,

where the core hole is defined as the absence of a core electron in a core level. This

process is essentially the photoelectric effect and forms the basis of three of the most

important spectroscopic techniques; x-ray photo-emission spectroscopy (XPS), x-ray ab-

sorption spectroscopy (XAS), and x-ray emission spectroscopy (XES) [3]. Due to these

advances, interest has grown in establishing an ideal theoretical model from which spectra

can be reliable simulated. Such models also allow further insight into the processes which

occur during spectroscopic measurements.

This report is primarily concerned with the theoretical simulation of XAS and RIXS.

Simulations will be carried out using two programs, CTM4XAS 5.2 and CTM4RIXS. The

transition metal (TM) compound, titanium dioxide (TiO2), will be the subject of each

investigation and will be compared against actual XAS and RIXS data of TiO2. For the

remainder of this introduction, one will explain the processes of XAS, XES, and RIXS,

including a brief description of TiO2. This is followed by a discussion, in chapter 2, on

the theoretical methods behind the spectroscopic simulations computed by CTM4XAS 5.2

and CTM4RIXS. Chapter 3. will describe the methods used to obtain simulated spectra,

after which, in chapter 4, one will present the simulated results obtained from CTM4XAS

5.2 and CTM4RIXS, drawing comparison to experimental data.

1

1.1. X-Ray Absorption Spectroscopy Chapter 1. Introduction

1.1 X-Ray Absorption Spectroscopy

In x-ray absorption spectroscopy, the absorption of x-rays by a sample is measured [3].

For instance, when an experimental sample is illuminated by synchrotron radiation, tran-

sitions of core electrons to the conduction band (CB) (or continuum) take place as they

absorb energy from the incident radiation.

Figure 1.1: Schematic diagram of x-ray absorption.

In Figure 1.1, a core electron has been excited from a core energy level (with binding

energy equal to εc, to an empty or unoccupied state with energy εw. Thus, the energy of

the incident photon ~Ω is such that:

~Ω = εw − εc (1.1)

This means that as light of intensity I0 is exposed to a sample, it is partially absorbed

resulting in an attenuated beam of intensity I. The amount of absorption is proportional

to the thickness (x) of the sample as well as the linear absorption coefficient, µ which

depends on the type of atoms and the density ρ of the material.

InI0/I = µx (1.2)

Plotting µ against the excitation energy will show features known as absorption edges

where the excitation energy corresponds to the difference between a core-hole binding

energy and an unoccupied state1. In other words, the photon energy is equal to the

binding energy of electrons in the shells (s, p, d, f,...) of the absorbing element.

1An XAS spectrum is also dependant upon the dipole selection rules. Refer to Appendix A, Section5.1.1.

2

1.2. X-Ray Emission Spectroscopy Chapter 1. Introduction

The absorption edges are labelled in the order of increasing energy:

K,L1, L2, L3,M1, ...

where the K edge relates to a 1s core level and where the L2,3 edges relate to a 2p core

level binding energy [3].

Thus, by knowing the energy of the incident X-ray, one can determine the energy

separation between the core level and the unoccupied level. This is how XAS probe the

local partial densities of unoccupied electronic states. Furthermore, because each element

has a characteristic binding energy for a certain core level, XAS is also an element specific

technique [4].

1.2 X-Ray Emission Spectroscopy

In XAS, the incident x-ray causes a core electron to move to an unoccupied state. This

electron leaves behind what is known as a core hole. X-ray emission spectroscopy de-

tects the emitted photon, which is created when an electron from either the valance or

conduction band refills the core hole created in XAS, as in Figure 1.2.

Figure 1.2: Schematic diagram of x-ray emission.

The energy of this emitted photon is equivalent to the energy difference between the

core level and valence level. This is how XES probe the local partial densities of occupied

electronic states. Thus, XES is a second-order optical process, where first a core electron

is excited by an incident x-ray photon (XAS) followed by the excited state decaying by

emitting an x-ray photon to fill the core hole [3]. As such, this makes XES a photon in,

photon out process, where no charging effect occurs, which makes this process favourable

to measure both conducting and insulating samples [4].

3

1.3. Resonant Inelastic X-Ray Spectroscopy Chapter 1. Introduction

Moreover, depending on the excitation energy, XES can be categorized even further.

When the core electron is excited to the high-energy continuum well above the absorption

threshold2, it is called non-resonant x-ray emission spectroscopy (NXES) [3]. NXES is

generally used to find out information about the partial density of occupied states.

If the core electron is resonantly excited to the absorption threshold by the incident

photon (as in the process of XAS), the resulting emission spectrum depends strongly on

the incident-photon energy Ω, and the process is referred to as resonant soft x-ray emis-

sion (RSXE). RSXE can either be resonant elastic x-ray scattering (RXES), or resonant

inelastic x-ray scattering (RIXS).

1.3 Resonant Inelastic X-Ray Spectroscopy

If the atomic state relaxes fully by the emission of a single photon of energy ω after having

been excited by the incident photon of energy Ω (as in Figure 1.2), then ω = Ω and one

says the photon was elastically scattered. If the atomic state relaxes fully by emitting

a photon of energy ω 6= Ω, due to a secondary process, then one says the photon was

inelastically scattered [5], as can been seen in Figure 1.3.

Figure 1.3: Schematic diagram of resonant inelastic x-ray scattering.

Elastic scattering is useful as a monitor of incident light and hence for calibration.

However, the inelastic scattering and the associated intermediate processes are generally

of more interest [5]. RIXS is useful for probing low energy excitations, such as d-d and

charge transfer excitations [4].

2This is the case for x-ray photoemission spectroscopy (XPS)

4

1.4. Titanium Dioxide Chapter 1. Introduction

Since RIXS includes absorption as an excitation process and further includes infor-

mation on radiative decay, the information given by it is much greater that that given

by the first-order optical processes of x-ray absorption or photo-emission. It provides

bulk-sensitive and site-selective information. However, the intensity of the signal in a

second-order optical process is much weaker than that of a first-order optical process

because the efficiency of x-ray emission is quite low. As a result, state-of-the-art experi-

mental instrumentation is needed to obtain precise experimental data [1].

1.4 Titanium Dioxide

Titanium dioxide (TiO2) is a transition metal compound that occurs in nature as the

minerals rutile, anatase and brookite. Rutile is one of the simplest and most common

structural types adopted by metal oxides and is the form that will used throughout this

report. It contains six coordinated titanium within a tetragonal lattice, which can be

thought of as a cubic lattice stretched along one of it’s lattice vectors. TiO2 has many

applications such as a photo-catalyst, sunscreen, electronic data storage (a memristor),

and pigment. Figure 1.4 depicts the structure of the rutile TiO2 unit cell from three

different perspectives.

Figure 1.4: (a) TiO2 rutile unit cell where stretched apex lies ∼ along the y-axis. (b)TiO2 rutile unit cell oriented such that the stretched apex lies in a vertical direction ∼along the z-axis. (c) TiO2 rutile unit cell as viewed when stretched apex lies ∼ along thex-axis.

5

Chapter 2

Theoretical Backgroud

In XAS, a core electron is excited to an empty state generating a core hole, for instance, a

2p core hole is created in the Ti4+ XAS transition 2p63d0 to 2p53d1. Thus, the final state

of the XAS process contains a partly filled core state (2p5) and a partly filled valence

state (3d1). A strong overlap between the 2p-hole and 3d-hole radial wave functions split

the XAS final states. These XAS final states created through splitting are known as

atomic multiplets. This overlapping of wave-functions also exists in the ground state, but

does not play an effective role here since the core states in the XAS ground state are

filled. This is why a single particle description is not capable of calculating the correct

density-of-states (DOS) for XAS of transition metal L2,3 edges.

Atomic multiplet calculations, where only the interactions within the absorbing atom

are considered, without any influence from the surrounding atoms will be described first.

However, in the case of 3d TM ions, atomic multiplet theory considered by it’s self, will

not adequately simulate XAS spectra as the effects of neighbouring ions in the 3d states

are too large. It turns out that both the symmetry effects and the charge-transfer effects

of the neighbours must be included explicitly. Thus, it is necessary to introduce the ligand

field multiplet model, which takes care of all the symmetry effects via the crystal field

effect. Followed by a description of charge transfer multiplet theory which allows the use

of more than one configuration so that ligand bonds can be accounted for [3]. Altogether,

these theories are used to theoretically calculate experimental spectra and to facilitate

the analysis of such data [4].

6

2.1. Atomic Multiplet Calculations Chapter 2. Theoretical Backgroud

2.1 Atomic Multiplet Calculations

Atomic multiplet calculations are based upon Hartree-Fock aproximations1 and atomic

multiplet theory [4]. Atomic multiplet theory is the description of atomic structure using

quantum mechanics [3]. The reason for employing quantum mechanics is due to the

nature of the elements within an atomic structure. In the classical description of an

electron orbiting a nucleus, the electron will accelerate and radiate energy. As a result,

the electron should eventually lose it’s energy and spiral into the nucleus, making atoms

quite unstable. However, this is not what is observed. Instead, atoms are stable on

time-scales much longer than predicted by the classical Larmor formula2 [6].

The starting point is the Schrodinger equation of a single electron in an atom:

HΨ = EΨ (2.1)

where H is the Hamiltonian operator equal to:

H =−~2

2m∇2 − Ze2

~r(2.2)

The solutions of this equation are the atomic orbitals as defined by their quantum

numbers n (principle), l (azimuthal), and m (magnetic). In atoms which are not influenced

by their surroundings and where more than one electron is present, there are two additional

terms in the atomic Hamiltonian:

1. The electron-electron repulsion (Hee) and

2. The spin-orbit coupling of each electron (H`s).

The total Hamiltonian is then give by:

HATOM =N∑i=1

~pi2

2m︸ ︷︷ ︸1

+

2︷ ︸︸ ︷N∑i=1

−Ze~ri

+∑pairs

e2

~rij︸ ︷︷ ︸3

+

4︷ ︸︸ ︷N∑i=1

ζ(~ri)li × si (2.3)

The first term represents the kinetic energy of the electrons, the second term represents

the electrostatic interaction of the electrons with the nucleus, the third term represents

the electron-electron repulsion (Hee), and the forth term stands for the spin-orbit coupling

(H`s) of each electron. The kinetic energy and the interaction with the nucleus are the

same for all electrons in a given atomic configuration [3]. Thus, the first two terms define

the average energy of the configuration (Havg) and do not contribute to the multiplet

1Refer to Appendix A, Section 5.1.2.2Refer to Appendix A, Section 5.1.3.

7

2.1. Atomic Multiplet Calculations Chapter 2. Theoretical Backgroud

splitting. The remaining two terms (Hee and H`s) define the relative energy of the different

terms within the configuration and contribute to the multiplet splitting [4].

However, Hee is too large to be treated as a perturbation and creates difficulty in

solving the Schrodinger equation. Thus, the central field approximation is applied [3].

For an electron pair where one electron is close to the nucleus and the other is far away,

one imagines that the radial component is the dominant one. This leads to the assumption

that, on average, a large fraction of the electrostatic electron-electron interaction can be

considered radial, in other words, represented by a central field. While the remaining non-

central part is small enough to be treated as a perturbation [7]. As such, the spherical

average of the electron-electron interaction 〈Hee〉 is separated from the non-spherical part

and added to Havg to form the average energy of the configuration. Thus, Havg does not

contribute to the multiplet splittings [8]. The modified electron-electron Hamiltonian H ′ee

where the spherical average has been subtracted is now small enough to be treated as a

perturbation [3].

H ′ee = Hee − 〈Hee〉 (2.4)

=∑pairs

e2

~rij〈∑pairs

e2

~rij〉 (2.5)

Therefore, H ′ee and Hls define the effective Hamiltonian responsible for the multiplet

splitting that determines the energy of the different terms in the configuration [4].

Atomic spectral lines correspond to transitions between quantum states of an atom.

These states are labelled by a set of quantum numbers summarized in what are known

as term symbols [9]. Thus, term symbols are an abbreviated description of the angular

momentum quantum numbers in a multi-electron atom [10]. They allow someone to

find out the three relevant quantum numbers as well as to determine the energy and

symmetry associated with a certain configuration. For instance, a 2p electron with L =

1, S = 1/2, and J = 1/2, 3/2, the term symbols are written as 2P1/2 and 2P3/2, where

spin-orbit coupling is important. These two terms are seen in the experimental 2p XAS

of a transition metal as the L2 and L3 peaks [4].

To determine the relative energies of different terms derived from the formula 2S+1XJ ,

one calculates the matrix elements of these states with the effective Hamiltonian:

Hee +H`z =∑pairs

e2

~rij+∑N

ζ(~ri)`i · si (2.6)

8

2.1. Atomic Multiplet Calculations Chapter 2. Theoretical Backgroud

The general formula to calculate the matrix elements of two-electron wave-functions

can be written as [3]:

〈2S+1LJ |e2

~r12|2S+1LJ〉 =

∑k

fkFk +

∑k

gkGk (2.7)

To obtain this equation, the radial parts F k and Gk have been separated using the

Wigner-Eckhart theorem3 and the Hamiltonian 1/ ~r12 has been expanded by Legendre

polynomials. For three or more electrons, it can be shown that the three-electron wave-

function can be built from the two-electron wave functions with the use of so-called

coefficients of fractional parentage [3].

F k and Gk in equation (2.7) are the Slater-Condon parameters for the direct Coulomb

repulsion and the Coulomb exchange interaction, respectively. Gk values have an approx-

imately constant ratio with respect to the F k values [3]. Within the program CTM4XAS

5.2, they are incorporated via the ‘Slater Integral Reduction (%)’ as Fdd, Fpd, and Gpd.

Fdd and Fpd represent the radial parts of the 2p-3d Coulomb multi-pole interactions whilst

Gpd represents the exchange multi-pole interactions [8]. The default setting, 1, for each

of these parameters within CTM4XAS 5.2, is equal to 80% of their calculated values [14].

This correction is to account for intra-atomic configuration interaction [8], or in other

words, the overestimation of the electron-electron repulsion in the atomic calculation [3].

For tetravalent titanium (Ti4+), a 3d0 → 2p53d1 transition occurs at the L2,3 edge

[14]. The initial state is given by the matrix element 〈3d1|HATOM |3d1〉 and the final state

by the matrix element 〈2p53d1|HATOM |2p53d1〉. The final state atomic energy matrix of

2p3d, consists of terms related to the two-electron Slater integrals (HELECTRO), and the

spin-orbit couplings of the 2p (HLS−2p) and 3d (HLS−3d) electrons [3]:

Heff = HELECTRO +HLS−2d +HLS−3d (2.8)

HELECTRO = 〈2p53d1| e2

~r12|2p53d1〉 (2.9)

HLS−2p = 〈2p|ζp`p · sp|2p〉 (2.10)

HLS−3d = 〈3d|ζd`d · sp|3d〉 (2.11)

The x-ray absorption transition matrix elements to be calculated are:

IXAS ∝ 〈3d0|p|2p53d1〉2 (2.12)

However, to theoretically simulate an x-ray absorption spectrum which lies in close

agreement to what is experimentally recorded, one needs to take into account solid state

effects. This is done by using the ligand field multiplet model which introduces crystal

3Refer to Appendix B.

9

2.2. Ligand Field Multiplet Theory Chapter 2. Theoretical Backgroud

field parameters into the calculation. After which, the charge transfer parameters are

then included into the calculation.

2.2 Ligand Field Multiplet Theory

The ligand field multiplet (LFM) model approximates a TM as an isolated atom sur-

rounded by a distribution of charges, which mimic the system, molecule, or solid, around

the TM [3]. Essentially it is applying a group theoretical treatment to the symmetry op-

erations present in common molecules [12]. The LFM Hamiltonian consists of the atomic

Hamiltonian, equation (3), to which a crystal field (CF) is added [3]:

HLFM = HATOM +HCF (2.13)

HCF = −eφ(~r) (2.14)

The crystal field consists of the electronic charge e times a potential that describes

the surroundings φ(~r). The potential φ(~r) is written as a series expansion of spherical

harmonics YLMs [3]:

ψ(~r) =∞∑

L=0

L∑M=−1

~rLALMYLM(Ψ, ψ) (2.15)

The crystal field is regarded as a perturbation, thus, the matrix elements of ψ(~r)

should be determined with respect to the 3d-orbitals [3].

2.3 Crystal Field Theory

Crystal field theory (CFT) is a model that describes the breaking of degeneracies of

electronic orbital states, usually d or f-orbitals, due to a static electric field produced by a

surrounding charge distribution. It is developed by considering the energy changes of the

five degenerate d-orbitals upon being surrounded by an array of point charges consisting

of the ligands. As a ligand approaches the metal ion, the electrons from the ligand will be

closer to some of the d-orbitals and farther away from others causing a loss of degeneracy.

The electrons in the d-orbitals and those in the ligand repel each other due to repulsion

between like charges. Thus, the d-electrons closer to the ligands will have a higher energy

than those further away which results in the d-orbitals splitting in energy [13].

Many systems possess a TM ion surrounded by six/eight ligands. The six ligands are

positioned on the three Cartesian axes, that is, on the six faces of a cube surrounding

the TM. They form the octahedral field. The eight ligands are positioned on the eight

corners of the cube and form the cubic field. Both these systems belong to the Oh point

10

2.3. Crystal Field Theory Chapter 2. Theoretical Backgroud

group which is a subgroup of the atomic SO3 [3]. If one were to rotate a featureless sphere

by any angle around any axis running through the sphere’s center, it will look exactly

the same. This is SO3 symmetry. Oh stands for full octahedral symmetry and is the full

symmetry group of the cube and octahedron.

To calculate an x-ray absorption spectrum in a cubic crystal field, the atomic transition

matrix elements must be branched from an SO3 (spherical) symmetry to an Oh (cubic)

symmetry [3]. The CTM4XAS 5.2 program implements this by running a program called

RAC2 which applies the necessary branching rules determined from group theory [14].

One can lower the symmetry even further from O3 symmetry to D4h (tetragonal) symmetry

by applying another set of branching rules, again determined from group theory. D4h is a

subgroup of Oh and belongs to the prismatic groups. In group theory terms, this can be

imagined as a coloured cube where two of the opposite faces have the same color whilst

all other faces have one different color. For TiO2, this translates as a stretching along one

of it’s lattice vectors, resulting in a stretched cube shape, as can be seen in Figure 2.1.

Figure 2.1: Rutile titanium oxide unit cell, where the stretched lattice vector is indicatedby the yellow line.

The radial parameters related to these branches are indicated as X40, X400, X420, and

X220 and can be related to other definitions such as 10 Dq, Ds, and Dt by comparing their

effects on the set of 3d functions. The most straightforward way to specify the strength of

the crystal field parameters is to calculate the energy separations of the 3d functions. In

Oh symmetry, there is only one crystal field parameter X40. This parameter is normalized

in a manner that creates unitary transformations in the calculations. Table 2.1 shows the

relation between the linear combination of X parameters to the linear combination of the

Dq, Ds, and Dt parameters, and the specific 3d orbitals they represent.

11

2.3. Crystal Field Theory Chapter 2. Theoretical Backgroud

Table 2.1: Energy of the 3d Orbitals Is Expressed In X400, X420, and X220 in the FirstColumn and in Dq, Ds, and Dt in the Second Coloumn [3].

Energy Expressed in X-Terms In D-Terms Orbitals

30−1/2 ·X400 − 42−1/2 ·X420 − 2.70−1/2 ·X220 6Dq + 2Ds - 1Dt x2 − y230−1/2 ·X400 + 42−1/2 ·X420 + 2.70−1/2 ·X220 6Dq - 2Ds - 6Dt z2

−2/3 · 30−1/2 ·X400 + 4/3 · 42−1/2 ·X420 − 2.70−1/2 ·X220 -4Dq + 2Ds - 1Dt xy−2/3 · 30−1/2 ·X400 − 2/3 · 42−1/2 ·X420 + 70−1/2 ·X220 -4Dq - 1Ds + 4Dt xz, yz

From this table, one can relate both notations and write X400, X420, and X220 as a

function of Dq, Ds, and Dt:

• X400 = 6.3012 ×Dq − 7

2× 30

12 ×Dt

• X420 = −52× 42

12 ×Dt

• X220 = −7012 ×Ds

In octahedral symmetry, the crystal field parameters split a 3d configuration into an eg

and a t2g configuration. t2g consists of the dxy, dxz, and dyz orbitals which will be lower in

energy than the eg configuration, consisting of the higher energy dz2 and dx2−y2 orbitals.

The reason the t2d group is lower in energy is because it is farther from the ligands than

the eg group and therefore experiences less repulsion [13]. In D4h symmetry, the t2g and

eg symmetry states split further into eg and b2g, respectively, and a1g and b1g. Depending

on the nature of the tetragonal distortion, either the eg or the b2g state have the lowest

energy [3].

Figure 2.2: The geometric shape of octahedral symmetry is represented on the left, whilstit’s crystal field splitting diagram is to the left

12

2.4. Charge Transfer Multiplet Theory Chapter 2. Theoretical Backgroud

2.4 Charge Transfer Multiplet Theory

Charge transfer effects are the effects of charge fluctuations in the initial and final states.

In atomic multiplet and LFM theories, a single configuration is used to describe the ground

state and final state. In the charge-transfer model, however, two or more configurations

are used. It does this by adding the configuration 3dn+1L to the 3dn ground state. In

the case of a TM oxide, in a 3dn+1L configuration an electron has been moved from the

oxygen 2p valence band to the metal 3d band. One can continue with this procedure and

add a 3dn+2L2 configurations, and so on. In many cases, two configurations are enough to

explain the spectral shapes of XAS. Unless one is dealing with a high valence state where

it can be important to include more configurations [3].

The charge-transfer effect adds a second dipole transition, a second initial state, and a

second final state. The two initial states and two final states are coupled by hybridization

(i.e. the charge-transfer effect between the ligand p and the TM 3d states). The mixing

Hamiltonian is given by

HMIX =∑v

V (Γ)(a+dvav + a+v adv) (2.16)

The XAS spectrum is calculated by solving the equations (2.17) and (2.18) below.

MI1,I2 = 〈3dn|HMIX |3dn+1L〉 (2.17)

MF1,F2 = 〈2p53dn+1|HMIX |2p53dn+2L〉 (2.18)

In the CTM4XAS 5.2 program, the charge transfer effects can be altered using a seven

different parameters. The first is the charge transfer energy, ∆, which gives the energy

difference between the (centers of the) 3dn and the 3dn+1L configurations. The effective

value of ∆ is affected by the multiplet and crystal field effects on each configuration. The

second is Udd which is the value of the on-site dd Coulomb repulsion energy. The third is

Upd which is the core hole potential [3]. In the case of XAS spectra, only the difference

between Upd and Udd is important. The remaining parameters, a1, b1, e, and b2 (the four

symmetries in tetragonal symmetry), are know as the hopping parameters. a1 (z2) and

b1 (x2 − y2) are part of the eg-orbitals. e (xz, yz) and b2 (xy) are part of the t2g-orbitals.

In Oh symmetry their values are a1 = b1 and e = b2 [14].

13

2.5. Resonant Inelastic X-Ray Scattering Chapter 2. Theoretical Backgroud

2.5 Resonant Inelastic X-Ray Scattering

By considering the entire process of RIXS as a set of atomic state transitions from ground

state |g〉 to an intermediate state |i〉 to a final state |f〉 of energies Eg, Ei, and Ef respec-

tively, on can define the Kramers-Heisenberg equation for the scattering cross section, σ,

as

σ(Ω, ω) ∝∑j

[∑i

〈j|T |i〉〈i|T |g〉Eg + Ω− Ei − iΓi

]2

× δ(Eg + Ω− Ej − ω) (2.19)

where T represents the radiative transition and Γi is the broadening due to the in-

termediate state core-hole lifetime [1]. The resonant behaviour arises at the absorption

threshold as the denominator terms Eg, Ω, and Ei, cancel leaving only the core-hole

broadening [1].

2.6 Spectroscopic Simulation Software

Throughout this project one used the programs CTM4XAS 5.2 and CTM4RIXS to carry

out all theoretical simulations. The source codes for the CTM4XAS 5.2 calculations

are the TT-MULTIPLET charge transfer multiplet programs, as have been originally

developed by Theo Thole and modified by Haruhiko Ogasawara. The Charge Transfer

multiplet program is a semi-empirical program that includes explicitly the important

interactions for the calculation of L-edge spectra. Interactions such as the core and

valence spin-orbit coupling, the core-valence overlap (multiplet effects), and the effects of

strong correlations within the charge transfer model [11].

CTM4XAS 5.2 calls and combines different programs to compute simulations. The

first program to be called is RCN2, which is used to compute the atomic parameters.

RCN2 can calculate any element and any transition in the periodic table. The only

information needed is the atomic number and the initial and final state configurations.

The Slater integrals (F2, G1, and G3) are scaled down to 80% of their ab initio values

[8]. If one would like the atomic multiplet spectrum to be calculated for the input of

RCN2, another programs called RCG2 is executed. The CTM4XAS 5.2 program then

runs RAC2, which calculates the crystal field multiplets using the output obtained from

RCN2. The reduced matrix elements of all necessary operators in the spherical group

are calculated with the use of Cowan’s atomic multiplet program. To obtain the reduced

matrix elements in any point group, Butler’s program is used for the calculation of all

necessary factors (3J symbols) [8]. The results of these two programs are written to matrix

files. These matrices are then used by an additional program called BAN2 to generate

the charge transfer multiplet spectrum [14].

14

Chapter 3

Computational Method

To simulate XAS spectra, the program ‘CTM4XAS 5.2’ was utilised. For RIXS simulation,

the program ‘CTM4RIXS’ was employed [11].

3.1 Method for Simulating XAS

CTM4XAS 5.2 was started by running the CTM4XAS52.exe file. This opened the pa-

rameter window, as can be seen in Figure 3.1.

Figure 3.1: CTM4XAS52 Parameter Window

15

3.1. Method for Simulating XAS Chapter 3. Computational Method

1. The first step was to define the configuration to be calculated. This was done by

clicking on the blue highlighted box located in the ‘Configuration and spectroscopy’

partition, beside ‘Electronic configuration’ in Figure 3.1. One then selected the

desired element and valence. This automatically filled the ‘Initial state’ and ‘Final

state’ boxes.

2. Next, the type of spectroscopy to be simulated was selected from the list of available

options. These were contained in the bordered area beside the blue highlighted

button in Figure 3.1. The orbitals to be probed were also selected here. Every XAS

simulation in this project was carried out with ‘2p’ selected.

3. The ‘Slater integral reduction (%)’ parameters Fdd, Fpd, and Gpd, were all left at

their default values of ‘1.0’ for the duration of the project.

4. The ‘SO coupling reduction (%)’ parameters ‘Core’ and ‘Valence’, were also left at

their default values of ‘1.0’ for the duration of the project.

5. The ‘Crystal field parameters (eV)’ partition contained options to define the type

of symmetry and crystal field one wished to be applied. The default setting was

Oh symmetry which only allowed the crystal field parameter ‘10 Dq’ to be defined.

When necessary, however, it was changed to other symmetries via the arrow button

beside it, which produced a drop down menu containing the other options such as

D4h symmetry, which was also used for calculations in this project.

6. When the symmetry was changed to D4h, the crystal field parameter ‘Dt’ and ‘Ds’,

in addition to ‘10 dq’, were made available. The desired values were then filled in

and the ‘Final state’ state box selected.

7. For charge-transfer calculation, one selected the box ‘CT’ to activate the ‘Charge

transfer parameters (eV)’ partition. The parameter ‘Delta’, ‘Udd’, and ‘Upd’ were

then changed to the desired values. The remaining parameters within this par-

tition, remained at their default values:T(eg)=2.0, T(eg)=2.0, T(t2g)=1.0, and

T(t2g)=1.0, for all simulations.

8. Within, the ‘Plotting’ partition’, the ‘Split’ button was selected and changed to

‘461’. The ‘Gaussian broadening’ was changed to ‘0.1’.The ‘Suppress sticks’ button

was de-selected. The remaining operations were left at their default values.

9. Finally, the button ‘Run’, located at the bottom of the screen, was selected, which

prompted one to name and save the calculation. After which, the software began a

series of calculations using the RCN2, RCG2, RAC2, BAN2, and PLO2 programs

described in chapter 2.

16

3.2. Method for Simulating RXIS Chapter 3. Computational Method

Although values for the Lorentzian broadening were defined in point 8. These were

improved upon manually, imposing a different broadening for each peak. This allowed

for a better fit to experiment. The details of the method used to do this can be found in

Appendix B, Section 5.321.

Simulations were plotted using Origin 8 where the output file from the PLO2.exe,

[filename].dat was imported as a single ASCII. Theoretical spectra were layered onto

already imported experimental spectra such that their energy values, along the x-axis,

aligned.

3.2 Method for Simulating RXIS

The first part of the RIXS calculation was carried out in the CTM4XAS 5.2 program

exactly as recounted in steps 1-9 for XAS calculations, except, instead of selecting the

2p XAS button, one chose the 2p-3d RIXS button. Once the calculation was complete,

CTM4RIXS was started by running the CTM4RIX.exe file. This opened the parameter

window. A picture of which can be found in Appendix B, Section 5.2.2.

1. First, one loaded the .ora files computed by the CTM4XAS 5.2 calculation for

absorption (in the first box) and for emission (in the second box). ‘Auto output’

and ‘Print energies and matrices’ were both selected. The ‘Create’ button was then

clicked and a pop out box appeared to which [1 5] was entered. One then clicked

OK and the program created the matrices.

2. The output automatically appeared in the next parameter box ‘AE matrix’, thus,

one clicked ‘Combine’. Another pop up box appeared where one chose the ground

level to be 1. One then clicked OK and the program combined the created matrices

from the first step.

3. Again, the output automatically appeared in the CB matric box. The RIXS button

was clicked and the results were automatically plotted within the program itself.

They were then extracted, and graphed with in Orgin.

17

Chapter 4

Results and Discussion

The overall aim of this project was to simulate the metal L-edge 2p-3d resonant inelastic

x-ray scattering spectrum of Ti3+, for comparison to a previously obtained experimental

RIXS spectrum. Moreover, one also investigated the relationship between the parameters

used for Ti4+ and Ti3+. This was done by examining whether the values used for pa-

rameters in Ti4+ calculations, could accurately simulate Ti3+ spectra. Simulations were

also compared to experimental data which acted as a reference to how simulated spectra

should appear, as well as displaying how characteristics within spectra were altered or

manipulated by different theoretical parameters.

The first section of results, deals with one’s simulation of Ti4+ 2p-3d XAS of TiO2

in octahedral and tetragonal symmetry. The second section, depicts calculations com-

puted for Ti3+ 2p-3d XAS, also in octahedral and tetragonal symmetry. The final section

presents one’s calculation of Ti3+ 2p-3d RIXS spectrum. For the Ti4+ calculations, I used

an experimental XAS rutile TiO2 Ti L-edge spectrum provided by Declan Cockburn [2]

for comparison. For the Ti3+ calculations, I used an experimental XAS sputtered rutile

TiO2 Ti L-edge spectrum provided by Stephan Callaghan [15] for comparison. The Ti3+

RIXS calculation is compared against experimental RIXS spectra of LAO3 SL excited

across the Ti3+ 2p3/2-3d eg threshold from a paper by Ke-Jin Zhou [16].

4.1 Simulated Ti4+ 2p XAS

I began by investigating how to simulate Ti4+ 2p XAS of TiO2. This was done in a step by

step process, where new parameters were introduced one by one to determine the effects

of each on spectra. Ti4+ was investigated first in octahedral symmetry (Oh) and then in

tetragonal symmetry (D4h). The parameters that were examined were the crystal field

relations and charge-transfer effects.

18

4.1. Simulated Ti4+ 2p XAS Chapter 4. Results and Discussion

4.1.1 Octahedral Symmetry

The first task was to find the best value of 10 Dq for a Ti4+ multiplet calculation in Oh

symmetry. In octahedral symmetry, 10 Dq is defined as the energy difference between the

teg states and the eg states, neglecting all atomic parameters [11]. Thus, the best fit 10

Dq was found by exploring how changing the value of 10 Dq affected the overall shape

of the spectrum. Examples of simulated spectra with varying values of 10 Dq can be

found in Appendix C, Section 5.3.1. Figure 4.1 below illustrates the simulated spectrum,

in red, of a Ti4+ 2p XAS calculation where 10 Dq=1.8 eV. This is compared against an

experimental XAS spectrum, in black, of a rutile TiO2 Ti L-edge.

Figure 4.1: The simulated XAS spectrum (in red) of a 2p Ti4+ calculation in Oh symmetrywith 10 Dq=1.8 eV, compared against an experimental XAS spectrum (in black) of rutileTiO2 Ti L-edge. The Lorentzian broadening values used for each peak in the simulatedspectrum were 0.1 eV, 0.6 eV, 0.5 eV, and 1.0 eV going from left to right respectively.

As one can see, in octahedral symmetry, the value of 10 Dq=1.8 eV compares nicely

against the overall shape of the experimental data. The crystal field parameter has split

the 2p3/2 (L3) and 2p1/2 (L2) into two more peaks know as teg and eg. Thus, four peaks

in total are present, in order of increasing energy known as the L3 t2g, L3 eg, L2 t2g and

L2 eg features.

19

4.1. Simulated Ti4+ 2p XAS Chapter 4. Results and Discussion

The crystal field parameter, 10 Dq, clearly has a large influence on the simulated XAS

Ti4+ spectrum. The other crystal field parameters Ds and Dt are not included as in

Oh symmetry as Ds=Dt=0. Hence, the next step was to record a simulated spectrum

including charge-transfer effects. Figure 4.2 shows the simulated spectrum of a Ti4+ 2p-

3d XAS in Oh symmetry including 10 Dq and charge-transfer effects ∆, Udd, and Upd

(obtained from Matsubara et al (2000) paper [17]), compared against an experimental

XAS Ti L-edge.

Figure 4.2: The simulated XAS spectrum (in red) of a 2p Ti4+ calculation in Oh symme-try with 10 Dq=1.7 eV, ∆=2.98 eV, Udd=4.0 eV, and Upd=6.0 eV, compared against anexperimental XAS spectrum (in black) of rutile TiO2 Ti L-edge. The Lorentzian broad-ening values used for each peak in the simulated spectrum were 0.1 eV, 0.6 eV, 0.5 eV,and 1.0 eV going from left to right respectively.

From Figure 4. the L3 t2g peak increases in intensity whereas the L3 and L2 eg peaks,

each slightly decrease in intensity. According to DeGroot [3], for systems with a positive

value of ∆, the main effects on the XAS spectral shape are, the formation of small satellites

and contraction of the multiplet structures. The latter is visible with regards to L3 and

L2 eg peaks, however, no satellites seem to be present. In any case, the formation of small

satellites or even the absence of visible satellite structures is a special feature of XAS.

This is because the local charge of the final state is equal to the charge of the initial state.

Implying that there is little screening and thus, few charge-transfer satellites [3].

20

4.1. Simulated Ti4+ 2p XAS Chapter 4. Results and Discussion

4.1.2 Tetragonal Symmetry

The Oh calculation in section 4.2 is a good start, however, more parameters need to

be incorporated if one is to achieve a simulation as close as possible to experimental

spectra. As was explained in section 2.2, the TiO2 rutile unit cell is not in fact a perfect

octahedral. Instead, it is distorted, where one of it’s lattice vectors is stretched, as can

be seen in Figure 2.1. This is due to the arrangement of oxygen ligands about the Ti4+

ion. Thus, due to these longer Ti-O bonds along only one of the axis, the point group

symmetry of the cation is reduced from Oh to D4h. Figure 4.3 displays the results of a

Ti4+ multiplet calculation in D4h symmetry.

Figure 4.3: The simulated XAS spectrum (in red) of a 2p Ti4+ calculation in D4h sym-metry with 10 Dq=2.8 eV, Dt=0.0771 eV, and Ds=0.1286 eV. These were derived fromthe experimental spectrum itself using a method outlined in Appendix C, Section 5.3.2.Compared against an experimental XAS spectrum (in black) of rutile TiO2 Ti L-edge.The Lorentzian broadening values used for each peak in the simulated spectrum were 0.1eV, 0.6 eV, 0.5 eV, and 1.0 eV going from left to right respectively.

The simulated XAS in D4h symmetry is definitely a better comparison of the experi-

mental XAS than the previous simulations in Oh symmetry. However, the splitting of the

eg state appears to be completely absent from the calculation, despite deducting values

for 10 Dq, Ds, and Dt from the experimental data. This I feel is down to the broadening

21

4.1. Simulated Ti4+ 2p XAS Chapter 4. Results and Discussion

used, perhaps the Lorentzian broadening of 0.6 eV, applied to the L3 eg feature was too

high. If I were to do this calculation again I would reduce this to 0.5 eV.

Next, was to explore the charge-transfer effects in D4h symmetry on Ti4+ XAS spectra.

Figure 4.4 shows the Ti4+ 2p-3d XAS calculation including crystal field and charge transfer

effects where were obtained from Matsubara et al (2000) paper [17], compared against an

experimental XAS Ti L-edge.

Figure 4.4: The simulated XAS spectrum (in red) of a 2p Ti4+ calculation in D4h sym-metry with 10 Dq=1.7 eV, Ds=0.9 eV, Dt=0.8 eV ∆=2.98 eV, Udd=4.0 eV, and Upd=6.0eV, compared against an experimental XAS spectrum (in black) of rutile TiO2 Ti L-edge.The Lorentzian broadening values used for each peak in the simulated spectrum were 0.1eV, 0.6 eV, 0.5 eV, and 1.0 eV going from left to right respectively.

Unlike the previous charge-transfer calculation in Oh symmetry, the t2g peaks appear

to have increased in absorption intensity, whereas the eg peaks have contracted. There

is also the appearance of small charge-transfer satellites, which were absent from the

previous Oh calculation. Another feature which appears, is a small initial peak in the L3

eg edge relating to the dz2 state which becomes degenerate in D4h symmetry. Although,

features relating to the parameters applied are present, the overall shape of the spectrum

is not a good comparison to experimental data. This could be because all the values for

the crystal field and charge-transfer were from Matsubara et al (2000) paper [17].

22

4.1. Simulated Ti4+ 2p XAS Chapter 4. Results and Discussion

Since I had already worked out the crystal field parameters in relation to the actual

experimental spectrum I was comparing my calculations against, I decided to repeat the

calculation using those crystal field values and broadenings, but kept the charge-transfer

values the same as the Matsubara et al (2000) paper [17]. Figure 4.5 shows the results.

Figure 4.5: The simulated XAS spectrum (in red) of a 2p Ti4+ calculation in Oh symmetrywith 10 Dq=2.8 eV, Ds=0.1286 eV, Dt=0.00771 eV ∆=2.98 eV, Udd=4.0 eV, and Upd=6.0eV, compared against an experimental XAS spectrum (in black) of rutile TiO2 Ti L-edge.The Lorentzian broadening values used for each peak in the simulated spectrum were0.175 eV, 0.4 eV, 0.45 eV, and 0.5 eV going from left to right respectively.

This concurs much better with the experimental XAS, and, although small, the dz2

peak is still present. Figure 4.5 shows just how important it is to chose the right broad-

enings.

23

4.2. Simulated Ti3+ 2p-3d XAS Chapter 4. Results and Discussion

4.2 Simulated Ti3+ 2p-3d XAS

Ti3+ 2p-3d XAS multiplet calculations were carried out for TiO2 to understand and to see

if one could simulate surface effects by using the same parameters used for Ti4+ applied

to a Ti3+ configuration. Examples of such surface effects include a missing oxygen and

reduced coordination at the surface, or surface reduction to Ti2O3, or an oxygen vacancy

resulting in the subsequent creation of Ti3+ ions surrounding the oxygen vacancy.

4.2.1 Octahedral Symmetry

The first calculation mimics that for Ti4+ in Oh symmetry except applied to Ti3+. Figure

4.6 displays the resulting spectrum in blue.

Figure 4.6: The simulated XAS spectrum (in red) of a 2p Ti3+ calculation in Oh symme-try with 10 Dq=1.8 eV, compared against an experimental XAS spectrum (in black) ofsputtered rutile TiO2 Ti L-edge. The Lorentzian broadening values used for each peak inthe simulated spectrum were 0.1 eV, 0.6 eV, 0.5 eV, and 1.0 eV going from left to rightrespectively.

Although, the L3 teg peak appears to correspond to the that of the sputtered experi-

mental spectrum, apart from that there appears to be little agreement between the two.

24

4.2. Simulated Ti3+ 2p-3d XAS Chapter 4. Results and Discussion

This maybe to be expected since the calculation is in Oh symmetry. If surface effects such

as oxygen vacancies etc. were present, the point-group symmetry would greatly reduced.

Figure 4.7 depicts the simulated spectrum of the same calculation as that for Ti4+ with

charge-transfer effects in Oh symmetry except applied to a Ti3+ configuration instead.

Figure 4.7: The simulated XAS spectrum (in red) of a 2p Ti3+ calculation in Oh symmetrywith 10 Dq=1.7 eV, ∆=2.98 eV, Udd=4.0 eV, and Upd=6.0 eV, compared against anexperimental XAS spectrum (in black) of sputtered rutile TiO2 Ti L-edge. The Lorentzianbroadening values used for each peak in the simulated spectrum were 0.1 eV, 0.6 eV, 0.5eV, and 1.0 eV going from left to right respectively.

At first glance, this calculation appears to disagree even more so with the sputtered

experimental XAS then the previous calculation without charge-transfer. However, the

L3 eg and L2 feature seem far to muted, and I expect the discrepancy to be due to one’s

choice of broadenings, the same used for Ti4+. Thus, one can concluded that so far, Ti4+

broadening cannot be applied to Ti3+ 2p-3d XAS multiplet calculations. To accurately

describe broadenings for Ti3+ one needs to adjust them according to the experimental

data.

25

4.2. Simulated Ti3+ 2p-3d XAS Chapter 4. Results and Discussion

4.2.2 Tetragonal Symmetry

Figure 4.8 is the Ti3+ 2p-3d TiO2 XAS multiplet calculation in d4h symmetry, using the

same values defined for the equivalent Ti4+ calculation, compared against the experimental

XAS of a sputtered rutile TiO2 sample.

Figure 4.8: The simulated XAS spectrum (in red) of a 2p Ti3+ calculation in D4h symmetrywith 10 Dq=2.8 eV, Dt=0.0771 eV, and Ds=0.1286 eV. Compared against an experimentalXAS spectrum (in black) of sputtered rutile TiO2 Ti L-edge. The Lorentzian broadeningvalues used for each peak in the simulated spectrum were 0.1 eV, 0.6 eV, 0.5 eV, and 1.0eV going from left to right respectively. The 10 Dq parameters in this calculation wereworked out from the experimental spectrum, the method of which is outlined in AppendixC, Section 5.3.2.

The simulated spectrum is quite reminiscent of that calculated in Oh Ti3+ with-

out charge-transfer effects and perhaps may fit better to the experimental data if the

Lorentzian broadenings were adjusted. If I were to repeat this calculation, in particular,

the broadenings applied to the L2 features would be lowered.

26

4.2. Simulated Ti3+ 2p-3d XAS Chapter 4. Results and Discussion

Finally, the Ti4+ calculation in D4h symmetry with charge-transfer effects was repeated

for the Ti3+ configuration. Figure 4.9 shows the Ti3+ 2p-3d XAS calculation including

crystal field and charge transfer effects where were obtained from Matsubara et al (2000)

paper [17], compared against an experimental XAS sputtered rutile Ti L-edge.

Figure 4.9: The simulated XAS spectrum (in red) of a 2p Ti3+ calculation in D4h symmetrywith 10 Dq=1.7 eV, Ds=0.9 eV, Dt=0.8 eV ∆=2.98 eV, Udd=4.0 eV, and Upd=6.0 eV,compared against an experimental XAS spectrum (in black) of rutile TiO2 Ti L-edge.The Lorentzian broadening values used for each peak in the simulated spectrum were 0.1eV, 0.6 eV, 0.5 eV, and 1.0 eV going from left to right respectively.

The bears very little resemblance to the experimental XAS of the sputtered TiO2

sample. One could adjust the broadenings, however, I feel even if broadenings were

adjusted it would still not adequately describe the experimental spectrum for the sputtered

rutile TiO2 sample. To amend this I feel that TiO6 values for the crystal field and charge-

transfer effects would be more appropriate. As well as this, if oxygen vacancies or surface

termination are properly taken into account, then there is no longer a cubic crystal field

since one of the ligand oxygens in the cubic octahedron is missing. It is surrounded by

only five oxygens in the cubic pyramid arrangement with the Ti in the middle of the

square. The local symmetry is then C4v. Therefore, to accurately simulate Ti3+ XAS

spectra, the point groups symmetry should be reduced to C4v symmetry accordingly.

27

4.3. Ti3+ 2p-3d RIXS Chapter 4. Results and Discussion

4.3 Ti3+ 2p-3d RIXS

A 2p-3d RIXS multiplet spectrum without charge transfer multiplets was calculated for

the Ti3+ ion in D4h symmetry. It was simulated using the assumption that it can be

described by the same crystal field parameters and point group symmetry used for XAS

Ti3+. Figure 4.10 contains the results.

Figure 4.10: The left graph contains a series of calculated RIXS spectra of TiO2 2p-3d inD4h symmetry with 10 Dq=2.65 eV, Ds=-0.1 eV, Dt=0.005 eV taken from [16]. Lorentzianbroadenings of 0.4 eV and 0.04 eV were applied. The graphs on the right are experimentaldata from Ke-Jin Zhou [16]. The top right graph is the total electron yield (TEY) mode2p-XAS spectra of LAO3 SL for both in-plane (black line) and out-of-plane (red line)polarization. The first two peaks are the Ti4+ 2p3/2-3d t2g and 2p3/2-3d eg resonances,respectively. Arrows label the incident energies used for RIXS. The lower right graph isa series of RIXS spectra of LAO3 SL excited across the Ti3+ 2p3/2-3d eg.

The simulated spectra compare to the experimental spectra quite well. Improvements

I would apply if I would be to fine tune the broadenings, lower the point group symmetry

to, and calculated the crystal field values from the experimental spectrum itself.

28

Chapter 5

Conclusions

Although the Ti4+ XAS simulations compared quite well against the XAS experimental

data of the Ti L-egde TiO2 sample, a concrete connection was not established between

the values of the parameters used in these Ti4+ calculations, to the parameters used in

the Ti3+ calculations. Instead, one needs to obtain separate values for Ti3+ simulations

to those of the Ti4+. My first, conclusion, would not only be to adjust the point group

symmetry to C4v, but, due to how strong the effect of the crystal field parameter 10 Dq

was on the Ti4+ spectrum, investigate the effects of different 10 Dq values for Ti3+.

However, in the RIXS simulation of Ti3+, I used crystal field parameters from Ke-Jin

Zhou [16]: Dq=2.65 eV, Ds=-0.1 eV, Dt=0.005 eV which compared well to the experi-

mental RIXS Ti3+. In some of the Ti4+ calculations in D4h symmetry I used: 10 Dq=2.8

eV, Dt=0.0771 eV, and Ds=0.1286 eV. It is interesting to note that there is not a large

difference between the 10 Dq values, however, not only is there a difference in magnitude

between the Dt values, but the Ds values are a different sign to each other. This suggests

that the connection between a Ti4+ calculation and a Ti3+ calculation could be in the

values for Ds and Dt.

The RIXS simulation was a good comparison to experiment. However, initially, one

had also wanted to include charge-transfer effects. Unfortunately, the CTM4XAS 5.2

software does not as yet allow for charge-transfer effects to be selected during a RIXS

calculation and no alternative programs were found. I feel this would be something

to compute in the future. As well as repeating the calculation in a lower point group

symmetry to replicate the distortions that exist within the Ti3+ unit cell.

All in all, the future of spectroscopy appears quite promising. With advances in

technology, it is more and more important to understand the surface properties of solids.

The transition metal compound TiO2 itself the subject of many investigations, resulting

in advances in technologies such as the memristor. The role of computational methods

for simulating spectroscopic experiments, therefore, is only going to grow as it becomes

increasingly more necessary to understand entirely the processes of spectroscopic methods.

29

Appendices

5.1 Appendix A

5.1.1 The Dipole Selection Rules

To a first approximation an XAS spectrum reflects the unoccupied density of states, albeit

modified by the selection of conduction band states due to the dipole or quadrupole nature

of the transition. To probe the 3d shell 2p, electrons are excited using the x-ray light where

the dipole selection rules ∆l = ±1, ∆j = 0,±1 allow the transitions [3]:

2p3/2 → 3d3/2.3d5/2, 4s

2p1/2 → 3d3/2, 4s

5.1.2 The Hartree-Fock Method

Hartree-Fock (HF) theory is one of the fundamental electronic structure theories. HF

theory assumes that the motion of each electron in a system can be described by a single-

particle wave function, i.e., no correlations to the instantaneous motion of other electrons

of the system are taken into account. It is a good approximation to the actual wave

function and thus considered as a useful starting point for more precise calculations.

5.1.3 Lamor Formula

When accelerating or decelerating, any charged particle (such as an electron) radiates away

energy in the form of electromagnetic waves. The Larmor formula is used to calculate the

total power radiated by a non-relativistic point charge as it accelerates.

P =e2a2

6πε0c3(SIunits) (5.1)

where e is the charge, a is the acceleration, and c is the speed of light.

30

5.2. Appendix B Chapter 5. Conclusions

5.2 Appendix B

5.2.1 Broadening Parameters

Each simulated spectrum within this project has been broadened by lifetime broadening

and experimental resolution. From detailed comparison with experiment, it turns out that

each of the four main lines has to be broadened differently [3]. However, although the

CTM4XAS 5.2 software allows one to define an experimental resolution using a Gaussian

broadening, it only allows two lifetime broadenings to be applied. To overcome this

limitation, one altered the .plo files produced from the calculation.

To do this, one opened up command prompt and edited the .plo file. Figure 5.1 shows

the .plo file before it has been edited. As one can see, only two Lorentzian broadenings

have been applied.

Figure 5.1: The original .plo file containing two Lorentzian and Gaussian broadeningparameters.

Thus, one added two more Lorentzian broadening lines stating the additional broad-

enings and the ranges to which they should apply. Figure 5.2 shows the edited .plo file.

31

5.2. Appendix B Chapter 5. Conclusions

Figure 5.2: The edited .plo file containing four Lorentzian and one Gaussian broadeningparameters.

Other changes which were also made, was the inclusion of the name of the .ora file to

which one wanted to apply broadening to. In Figure 5.2 have included this in a yellow box.

This is because .ora files are only suitable for calculation not containing charge-transfer

effects. If one is editing a calculation for charge-transfer effects this needs to be changed

to .oba for the new altered .plo file to work. I also changed the names of the .dat files in

the .plo file to the same name as the .ora/.oba file.

Once alterations had been made, the new file was saved and one returned to command

prompt. Although the .plo file now contains the new broadenings they have yet to be

applied to the calculations. To do this, one ran PLO1.exe with the edited .plo file. Figure

5.3 depicts the action of this.

32

5.2. Appendix B Chapter 5. Conclusions

Figure 5.3: Command prompt running the PLO1.exe for the editted .plo file with theresults printed below.

To display the altered spectra, the new .dat files were just imported into Origin and

graphed.

33

5.2. Appendix B Chapter 5. Conclusions

CTM4RIXS Paramter Window

Figure 5.4: CTM4RIXS Parameter Window34

5.3. Appendix C Chapter 5. Conclusions

5.3 Appendix C

5.3.1 Varying the Crystal Field

As one already mentioned, to find the best value of 10 Dq for a Ti4+ multiplet calculation

in Oh symmetry, one explored how changing the 10 Dq value affected the overall shape of

the spectrum. The first value to be considered was 10 Dq=0. I chose this as I wanted to

see how the simulated XAS spectrum of the Ti4+ calculation appeared before any crystal

field affects were included. Figure 5. shows the result.

Figure 5.5: The simulated XAS spectrum of a 2p Ti4+ calculation in Oh symmetry with10 Dq=0 eV.

Only the L3 and L2 features are present, respectively. This is to be expected as in

octahedral symmetry, 10 Dq is defined as the energy difference between the teg states and

the eg states, neglecting all atomic parameters.

10 Dq was then varied between -2.0 eV and +2.0 eV in steps of 0.4 eV, to see how

negative values compare against positive values.

35

5.3. Appendix C Chapter 5. Conclusions

Figure 5.6: The simulated XAS spectrum of a 2p Ti4+ calculation in Oh symmetry with10 Dq varied between -2.0 eV and +2.0 eV in steps of 0.4 eV.

The main difference appears to be the slight shift of the L3 and L2 t2g states of the

negative calculations to the left. Creating a smaller separation between the L3 and L2

features. Although, the difference is quite small, from consulting values used within

similar calculations contained in the literature pertaining to TiO2, a positive 10 Dq value

appears to be the best match. I therefore calculated spectra with 10 Dq varying between

1 eV and 3 eV in steps of 0.25 eV. Figure 5. shows the results.

36

5.3. Appendix C Chapter 5. Conclusions

Figure 5.7: The simulated XAS spectrum of a 2p Ti4+ calculation in Oh symmetry with10 Dq varied between 1.0 eV and 3.0 eV in steps of 0.25 eV.

Continuing this process, one eventually decided upon a value of 10 Dq=1.8 eV, as

this appeared to replicate the crystal field found in experimental XAS TiO2 Ti L−edge

spectra the best.

37

5.3. Appendix C Chapter 5. Conclusions

5.3.2 Crystal Field Parameters

For calculations involving more than just the 10 Dq crystal field value, i.e., D4h symmetry

calculations, I deducted 10 Dq, Dt, and Ds from Declan’s experimental spectra of TiO2

rutile Ti L-edge.

Figure 5.8: The energy splitting of Oh symmetry and D4h symmetry.

From Figure 5 :

∆ = (dx2 − y2)− (dxy) (5.2)

= 10Dq (5.3)

From Table 2.1 :

(dxy, dyz)− (dxy) = (−4Dq − 1Ds+ 4Dt)− (−4Dq + 2Ds− 1Dt) (5.4)

= −3Ds+ 5Dt (5.5)

and

(dx2 − y2)− (dz2) = (6Dq + 2Ds− 1Dt)− (6Dq − 2Ds− 6Dt) (5.6)

= 4Ds+ 5Dt (5.7)

Assuming there is no degeneracy of the dxy, dyz, and dxz states, one sets −3Ds +

5Dt = 0. However, there appears to be a slight splitting of the L3 eg peak in the

experimental Ti L−edge XAS. The higher energy peak corresponds to dz2 and the lower

to dx2− y2. Thus, by determining the difference in energy between these two peaks from

38

5.3. Appendix C Chapter 5. Conclusions

the spectrum’s graph I equated 4Ds + 5Dt = 0.9eV . Values for Ds and Dt were then

found by simultaneous equations.

−3Ds+ 5Dt = 0eV (5.8)

4Ds+ 5Dt = 0.9eV (5.9)

−7Ds = 0.9eV (5.10)

Ds = −0.9eV/7 (5.11)

Ds = −0.1286 (5.12)

The value for Ds was then substituted into −3Ds+ 5Dt = 0eV and Dt was found to

be Dt=-0.0771 eV. The 10 Dq values was also measured from the spectrum and found to

be 10 Dq =2.8 eV.

39

Chapter 5. Conclusions

5.3.3 RIXS Simulation: Incident Energy Vs Emitted Energy

Figure 5.9: RIXS Simulation Calculation Incident Energy Versus Emitted Energy Plot.

40

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42